We recently discovered an error in the implementation of the comparison with the theory of Einarsson et al. (Reference Einarsson, Candelier, Lundell, Angilella and Mehlig2015a ) that was performed in the paper of Di Giusto et al. (Reference Di Giusto, Bergougnoux, Marchioli and Guazzelli2024). This error does not invalidate the experimental work or the main conclusions of Di Giusto et al. (Reference Di Giusto, Bergougnoux, Marchioli and Guazzelli2024). It only affects the theory calculations and therefore its comparison with the experimental results. It does not affect the comparison with the theory of Dabade et al. (Reference Dabade, Marath and Subramanian2016).
In the paper of Di Giusto et al. (Reference Di Giusto, Bergougnoux, Marchioli and Guazzelli2024), the theory of Einarsson et al. (Reference Einarsson, Candelier, Lundell, Angilella and Mehlig2015a
) is introduced in Appendix B.1. The coefficients
${\beta _i}$
of the dimensional equation (B1) should have unit of time. However, these coefficients were given in dimensionless form (obtained using the shear rate) in figure 2 of Einarsson et al. (Reference Einarsson, Candelier, Lundell, Angilella and Mehlig2015b
) which is reproduced in figure 14 of Di Giusto et al. (Reference Di Giusto, Bergougnoux, Marchioli and Guazzelli2024). The error arises from using the dimensionless coefficients instead of the dimensional ones in equation (B1).
Correcting this error requires modifying figures 9, 10 and 11 of Di Giusto et al. (Reference Di Giusto, Bergougnoux, Marchioli and Guazzelli2024) as well as the corresponding JFM notebooks. The new figures (see figures 1, 2 and 3) demonstrate a closer match between experimental results and theoretical predictions. This is particularly evident in the fibre dynamics, where good agreement is maintained even for a particle Reynolds number of order
$O(1)$
. Discrepancies are more pronounced for oblate bodies and when moving away from the theory validity limit, particularly for larger Reynolds numbers.
Figure 1. Evolution of the components of the orientation vector
${\boldsymbol{n}}$
, displayed as vertically aligned panels for 3 typical runs against the dimensionless time
$t\dot \gamma $
, for the fibre CYL10 with aspect ratio
$r = 9$
and confinement ratio
$\kappa = 0.19$
: (a)
$R{e_p} = 0.15$
; (b)
$R{e_p} = 1.0$
. Comparison with the theory of Einarsson et al. (Reference Einarsson, Candelier, Lundell, Angilella and Mehlig2015a
), presented in
${\rm{\S}}$
B.1 is also given as black dashed lines. See Supplementary Materials for the directory of the figure including the data and the Jupyter notebook.
Figure 2. Evolution of the components of the orientation vector
${\boldsymbol{n}}$
, displayed as vertically aligned panels for 3 typical runs against the dimensionless time
$t\dot \gamma $
, for the disk CYL005 with aspect ratio
$r = 0.05$
and confinement ratio
$\kappa = 0.19$
: (a)
$R{e_p} = 0.24$
; (b)
$R{e_p} = 0.8$
. Comparison with the theory of Einarsson et al. (Reference Einarsson, Candelier, Lundell, Angilella and Mehlig2015a
), presented in
${\rm{\S}}$
,B.1 is also given as black dashed lines. See Supplementary Materials for the directory of the figure including the data and the Jupyter notebook.
Figure 3. Evolution of the components of the orientation vector
${\boldsymbol{n}}$
, displayed as vertically aligned panels for 3 typical runs against the dimensionless time
$t\dot \gamma $
, for the oblate spheroid ELL06 with aspect ratio
$r = 0.6$
and confinement ratio
$\kappa = 0.17$
at particle Reynolds number
$R{e_p} = 0.43$
. Comparison with the theory of Einarsson et al. (Reference Einarsson, Candelier, Lundell, Angilella and Mehlig2015a
), presented in
${\rm{\S}}$
,B.1 is also given as black dashed lines. See Supplementary Materials for the directory of the figure including the data and the Jupyter notebook.
The intriguing discrepancy between the experiments and the predictions of Einarsson et al. (Reference Einarsson, Candelier, Lundell, Angilella and Mehlig2015b ) and Dabade et al. (Reference Dabade, Marath and Subramanian2016) regarding the stability of the tumbling orbit still remains. A joint effort is underway with Z. Wang, X. de Wit, B. Mehlig and F. Toschi to analyse the origin of these discrepancies, and a future paper will be dedicated to this analysis.
Note that the correct dimensional coefficients
${\beta _i}$
have been used in the subsequent paper of Di Giusto et al. (Reference Di Giusto, Bergougnoux and Guazzelli2025) in the comparison with the theory of Einarsson et al. (Reference Einarsson, Candelier, Lundell, Angilella and Mehlig2015a
) for rings and disks.
We also use the opportunity of this corrigendum to correct figure 6(a) and (b) of Di Giusto et al. (Reference Di Giusto, Bergougnoux, Marchioli and Guazzelli2024) as well as the movies corresponding to figure 6(c), (d), (e), and (f), since the shear was mistakenly inverted, see the new figure 4 and the new movies.
Figure 4. Experimental Jeffery orbits at two Reynolds numbers for the fibre CYL10 (top-row panels), the spheroid ELL06 (middle-row panels) and the disk CYL01 (bottom-row panels): (a) Fibre,
$r = 9.0$
,
$R{e_p} = 0.08$
; (b) Fibre,
$r = 9.0$
,
$R{e_p} = 1.0$
; (c) Spheroid,
$r = 0.6$
,
$R{e_p} = 0.02$
; (d) Spheroid,
$r = 0.6$
,
$R{e_p} = 0.43$
; (e) Disk,
$r = 0.1$
,
$R{e_p} = 0.05$
; (f) Disk,
$r = 0.1$
,
$R{e_p} = 1.32$
. The particles considered in this figure are shown in the vorticity-aligned position with their orientation vector
${\boldsymbol{n}}$
highlighted in cyan. The coloured dots represent the intersections of the axis given by the orientation vector
${\boldsymbol{n}}$
with the half sphere of radius
$\ell $
for the prolate particles and
$a$
for the oblate particles, respectively. The corresponding Jeffery orbits are also displayed as solid black lines and were obtained by integrating equation (1.1) from an initial condition given by the first flow-aligned orientation of each experiment. See Supplementary Materials for animations.
Acknowledgements
We would like to thank Z. Wang, X. de Wit, B. Mehlig and F. Toschi for identifying the error in our calculation of the theory of Einarsson et al. (Reference Einarsson, Candelier, Lundell, Angilella and Mehlig2015a ). We would also like to thank A. Joshi and D. Koch for identifying a rotation in the opposite direction to that expected in the data of Figure 6.
Declaration of interests
The authors report no conflict of interest.
